Article Asymmetric Damage Segregation Constitutes an Emergent Population-Level Stress Response Graphical Abstract Highlights d Lineages and growth rate distributions reveal effects of ADS d Asymmetry of segregation is dependent on single-cell damage d The population benefit of ADS increases with stress Authors Søren Vedel, Harry Nunns, Andrej Kosmrlj, Szabolcs Semsey, Ala Trusina Correspondence [email protected] (S.V.), [email protected] (A.T.) In Brief Asymmetric damage segregation (ADS) is non-random, asymmetric partitioning of damage at cell division. Using theory and time-lapse imaging of E. coli colonies, we find that the extent of ADS increases with severity of stress through a combination of single-cell and population-level feedbacks. Vedel et al., 2016, Cell Systems 3, 187–198 August 24, 2016 ª 2016 Elsevier Inc. http://dx.doi.org/10.1016/j.cels.2016.06.008
Asymmetric Damage Segregation Constitutesan Emergent Population-Level Stress ResponseSøren Vedel,1,2,* Harry Nunns,1,3 Andrej Ko�smrlj,4 Szabolcs Semsey,1 and Ala Trusina1,*1Center for Models of Life2Niels Bohr International Academy
Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark3Division of Biology and Biological Engineering, California Institute of Technology, 1200 E. California Boulevard, Pasadena, CA 91125, USA4Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Asymmetric damage segregation (ADS) is a mecha-nism for increasing population fitness through non-random, asymmetric partitioning of damagedmacro-molecules at cell division. ADS has been reportedacross multiple organisms, though the measuredeffects on fitness of individuals are often small.Here, we introduce a cell-lineage-based frameworkthat quantifies the population-wide effects of ADSand then verify our results experimentally in E. coliunder heat and antibiotic stress. Using an experi-mentally validated mathematical model, we findthat the beneficial effect of ADS increases withstress. In effect, low-damage subpopulations dividefaster and amplify within the population acting likea positive feedback loop whose strength scaleswith stress. Analysis of protein aggregates showsthat the degree of asymmetric inheritance is damagedependent in single cells. Together our results indi-cate that, despite small effects in single cell, ADS ex-erts a strong beneficial effect on the population leveland arises from the redistribution of damage withina population, through both single-cell and popula-tion-level feedback.
INTRODUCTION
To cope with unfavorable environmental conditions, cells have
evolved myriad strategies that operate on both the single-cell
and population level. As a first line of defense, cells use versatile
cellular repair mechanisms to counteract macromolecular dam-
age caused by environmental stress. However, the repair is
not perfect andmay not prevent accumulation of DNAmutations
or protein aggregates within single cells. To accommodate this,
additional stress responses operate at the population level within
populations of genetically identical individuals. In particular, dur-
ing cellular division in bacteria (Winkler et al., 2010; Stewart et al.,
2005), yeasts (Coelho et al., 2013; Aguilaniu et al., 2003; Zhou
et al., 2011; Liu et al., 2011), and stem cells (Fuentealba et al.,
2008; Bufalino et al., 2013; Rujano et al., 2006), accumulated
C
damage is unequally allotted to the two resulting cells. This
asymmetric damage segregation (ADS) produces one faster-
growing ‘‘rejuvenated’’ cell at the cost of a more damaged,
slower-growing ‘‘aged’’ cell. In silico models suggest that ADS
increases the reproductive rate and survival of cellular popula-
tions (Chao, 2010; Ackermann et al., 2007; Erjavec et al., 2008;
Watve et al., 2006) under the conditions when efficiency of repair
is low compared to the rate of damage accumulation (Clegg
et al., 2014). Despite these potential benefits, ADS is not an obli-
gate response and is observed in only a fraction of cells (Yennek
et al., 2014), cell types (Bufalino et al., 2013), and environments
(Stewart et al., 2005; Wang et al., 2010; Coelho et al., 2013; Yen-
nek et al., 2014; Rang et al., 2012).
While the single-cell mechanisms underlying ADS are partially
understood (Spokoini et al., 2012; Liu et al., 2011; Malinovska
et al., 2012; Kaganovich et al., 2008; Specht et al., 2011; Winkler
et al., 2010), it remains unclear what the key determinants are
for ADS to emerge and become beneficial at the level of the
population. To characterize the extent of asymmetry, many
studies focus on pairwise comparisons of cells after division
(Coelho et al., 2013; Stewart et al., 2005; Lindner et al., 2008;
Winkler et al., 2010; Ackermann et al., 2007; Bufalino et al.,
2013; Yennek et al., 2014). This approach may underestimate
population-level effects as it relies on the assumptions that all
cell pairs exhibit the same level of ADS. To our knowledge, thor-
ough analysis of the actual population structure has been
performed. Moreover, it is unknown how significant the effects
of ADS are when compared to other sources of cell-to-cell
heterogeneity.
Here, we address these issues using asymmetric division of
protein aggregates in Escherichia coli as a model system. In
E. coli, damaged proteins form aggregates throughout the cell;
these localize to the older pole (pre-existing from previous
division) after several cellular divisions (Lindner et al., 2008).
By combining whole-population experiments—where single-
cell growth and aggregate dynamics were measured for over
11,000 E. coli cells—with mathematical model, we find that the
magnitude of ADS within a population is an emergent property
that scales with the level of stress applied. The emergence is
mediated by single-cell and population-level feedback loops,
acting to continuously re-distribute damage within individuals
and optimize population-level growth. Contrary to earlier as-
sumptions, this comprehensive whole-population mapping re-
veals that the levels of ADS are not the same in all cells but are
ell Systems 3, 187–198, August 24, 2016 ª 2016 Elsevier Inc. 187
(A) Schematic of the damage segregation in a lineage-based model. Old poles are marked in magenta and new poles in green; cell pole age is measured by its
number of consecutive old-pole divisions. Blue stars mark inherited damage; orange stars mark new damage acquired between two consecutive divisions. Right
panel summarizes key equations of the mathematical model.
(B) Mean doubling time hTi increases with stress, l, but decreases with asymmetry, a. Circles, simulation; dashed lines, theoretical expression (Equation S7). We
use a stochastic simulation algorithm for computational efficiency (see Experimental Procedures), and the error bars indicate the SEM caused by the
stochasticity.
(C) The population benefit B from ADS increases with stress and asymmetry. The benefits for doubling times, BhTi and damage BhDi, are defined as fold decrease
in mean doubling time and damage due to ADS (see Experimental Procedures). Error bars indicate the SE of the benefit due to the use of a stochastic algorithm.
(D and E) Both the asymmetry and stress affect the shape of the doubling-time distribution by increasing the cell-to-cell variability, which is quantified by
distribution width, s, in (E). Dashed lines show theoretical results of the deterministic model (Equation S8); filled circles mark the distribution widths from
simulations in (D).
See also Figure S1.
determined by the cell’s lineage history, such that cells withmore
damage exhibit a higher level of asymmetry. Finally, using line-
age-based tools for ADS analysis that we introduce here, we
show that measured levels of ADS indeed confer a population-
level advantage even when masked by cell-to-cell variability
that is unrelated to ADS.
RESULTS
Predictions of the Population Benefit from ADS Using aLineage-Based Mathematical ModelTo investigate how ADS affects population structure and dy-
namics, we considered a conceptually simple mathematical
model of cells in growing population subject to ADS (Figure 1A).
188 Cell Systems 3, 187–198, August 24, 2016
Many molecular details regarding ADS remain obscure, or cell-
or damage-type specific, so we focused instead on simple
mathematical implementations of general concepts relevant to
ADS. While similar in silico models were investigated earlier
(Ackermann et al., 2007; Erjavec et al., 2008; Chao, 2010), the
effects of environmental stress nonetheless remain largely un-
explored. By treating the population as a set of evolving lineages
rather than a collection of individual cells (see Supplemental In-
formation, section V), we derived analytical relationships be-
tween (1) the severity of environmental stress l, (2) physiological
properties such as doubling times T, damages D, and (3) and
cell-to-cell variation s, representing the population effects of
ADS (see Figure 1 and Equations S7 and S8). Theoretical results
were paralleled by numerical simulation of the model (see
Experimental Procedures and Supplemental Information, sec-
tion IV).
In thismodel, each cell inherits a lineage-dependent amount of
accumulated damage, Dinh, from its ancestor (Figure 1A). Over
its lifetime between divisions, each cell acquires additional dam-
age, proportional to the severity of stress l, resulting in the total
damage
D=Dinh + l (Equation 1)
prior to division. Accumulated damage slows down cell growth
(Lindner et al., 2008), so we set the doubling time T of the cell
to linearly depend on its total amount of damage,
T = tmin +mD= tmin +mðDinh + lÞ; (Equation 2)
where m is a proportionality constant relating damage and time,
and tmin is the minimum doubling time of a cell set by mecha-
nisms unrelated to ADS. ADS is introduced into the model
through an asymmetry parameter a ð0%a%1Þ that dictates the
amount of damage inherited by each sister cell at division. The
old-pole (magenta, Figure 1A) cell inherits a larger fraction of
the ancestor’s damage Doldinh = ð1+ aÞ=2Dancestor , while the new-
pole cell inherits a fraction Dnewinh = ð1� aÞ=2Dancestor , resulting in
a doubling-time difference between two sister cells
Told � Tnew =maDancestor : (Equation 3)
This equation reveals that the difference in sister-cell doubling
times is a product of both the lineage history (via Dancestor ) and
single-cell behavior (via a). We initially treat the asymmetry
parameter as a constant across all cells, but we will later relax
this assumption. To speed up the convergence of our simula-
tions, we have added small zero-mean noise to the cell’s
doubling time. Theoretical results are deterministic (see Experi-
mental Procedures).
In linewithprevious insilicofindings (Ackermannetal., 2007;Er-
javec et al., 2008; Chao, 2010), our modeling results show that
ADS reduces the mean doubling time. We find that this effect is
greater under higher stress (Figures 1B and S1A). Consequently,
the benefit of ADS (defined as the fold-change improvement of
physiological properties at some asymmetry ðaR0Þ relative to
starting from single cell and up to �300 cells. Since protein ag-
gregates are known to be the predominant factor in E. coli
ADS (Lindner et al., 2008; Winkler et al., 2010), we expressed
the heat shock chaperone ClpB fused to YFP via a medium-
copy plasmid within clpB knockout cells (see Experimental
Procedures and Figure 4A). This reporter allowed us to experi-
mentally measure the asymmetry parameter a by monitoring
aggregate formation and inheritance. Compared to other chap-
erones, ClpB showed a higher level of co-localization with aggre-
gates, and the fusion has been shown to be functional (Winkler
et al., 2010), making it an effective marker. We also verified the
results of experiments with wild-type E. coli (all experiments
are summarized in Table S1).
Lineage-Based Data Analysis Captures the Population-Level Effects of ADSWedeveloped a lineage-baseddata-analysis framework centered
on binning cells with similar lineage histories to properly investi-
gate the population structure. If the damage re-distribution is
governed primarily by ADS, cells in the same bin would have
similar damage levels. Variations between bins (caused by differ-
ences in accumulated damage) is therefore a signature of
ADS, while within-bin variations would be caused by ADS-unre-
lated variations, which we refer to as noise. We approximated
the full lineages by binning the cells based on the number h
of consecutive events of damage accumulation (positive h) or
evasion (negative h) for both the current cell and its immediate
ancestor: hcurrent, hancestor (see Experimental Procedures and
Figure 2B). The median doubling time in each lineage served
as our noise-filtered readout coupling to the lineage-specific
Cell Systems 3, 187–198, August 24, 2016 189
A B
DC
Figure 2. Lineage-Based Approach to Quantify Population Effect of ADS
(A) Each lineage represents a unique history of damage-accumulation (+)/evasion (�) events and therefore results in different doubling timeswithin the population.
The extreme, damage-accumulating (all magenta) and damage-evading (all green) lineages are shown.
(B) Lineage groups are created by pooling together lineages with similar histories of damage redistribution events. The median doubling times of the each lineage
group (dash-dotted lines) are used as a proxy of the noise-filtered doubling time.
(C) The heatmap gives pairwise differences in median doubling times for each pair of lineage groups. We approximate the history of the lineage by a pair of
numbers ‘‘hancestor, hcurrent’’ where hancestor is the number of consecutive damage-evasion (�)/accumulation (+) events for the ancestor and hcurrent is the same
quantity calculated for the current cell. The entries are symmetric around the diagonal, with a difference only in sign.
(D) Distributions of noise-filtered doubling times. Each value underlying the distribution represents the median doubling time of a lineage group and carries a
weight corresponding to the fraction of the population in this group. Null distributions reflect ADS-unrelated variations due to finite sample size and are obtained
separately at each stress level by random reshuffling of single-cell doubling times between the lineage groups.
See also Figure S2.
accumulated damage, and we confirmed that the statistical prop-
erties of the noise were similar across all lineages. Including all lin-
eages of the population in our analysis (Figure 2B), we obtained a
noise-filtered population distribution of doubling times (Figure 2D).
190 Cell Systems 3, 187–198, August 24, 2016
To test this approach,we re-analyzed the low-stressdatasampled
from a different strain (MG1655) by Stewart et al. (2005). The line-
age-based analysis applied to this vast dataset, which consists of
over90colonies, capturessignificantdifferences indoubling times
between lineages differing by only one damage-accumulation/
evasion event (Figure S2).
Heat Shock Experiments Confirm Predicted Stress-Adaptive Population Response from ADSWe compared the growth of MC4100DclpB E. coli expressing
ClpB-YFP under minimal stress at 37�C to the heat stress at
42�C. Our data-analysis methods revealed that the doubling
time of a cell is indeed highly dependent on its lineage history.
We quantified this by the difference in median doubling times
between pairs of lineages in Figures 2C, 3A, 3B, and S2. As ex-
pected from our model, pairs of lineages with more consecutive
damage-accumulating/evading events were characterized by
larger time differences. Also, while doubling times did converge
to a finite value after successive rounds of damage accumulation
(or damage evasion) (Figure S3B), most lineages yielded
doubling times that were between the two extremes (Figures
3C and 3D). These observations are in agreement with themodel
(Figure 1D) and provide experimental confirmation that ADS
diversifies cells into numerous subpopulations, and not two
subpopulations as suggested earlier (Wang et al., 2010).
Next, we investigated the effects of stress on the distribution
of doubling times. We found that the mean doubling time from
the noise-filtered data increased for cells exposed to heat stress
(from 26.9 at 37�C min to 28.8 min at 42�C, p < 0.01 (bootstrap-
ping, see Experimental Procedures), Figures 3C and 3D), which
is in agreement with themodel (Figure 1B and Equation 2). More-
over, we found that the population diversity (quantified by the
width s of the doubling-time distributions in Figures 3C and 3D
and heatmaps in Figures 3A and 3B) is nonzero at low stress (sta-
tistically significant compared to the null distribution, see Exper-
imental Procedures) and furthermore increased under the heat
stress (from s37 = 1.1 min to s42 = 2.0, p < 0.01, bootstrapping;
see Experimental Procedures and Figure S3A). These findings
confirm the stress adaptation predicted by the model.
Taken together, these observations confirm the key model
predictions about the population-level effects of ADS. Our re-
sults also illustrate that, while ADS is not absent under low levels
of stress, it becomesmore pronounced at higher stresses. These
findings appear to be general, as we have found qualitatively
similar results in repeat experiments with antibiotic stress in
MC4100DclpB + ClpB-YFP E. coli (0.5 mg/ml kanamycin) (Fig-
ures S3C–S3F) and in the parental MC4100 strain under both
types of stress (Figures S3G–S3P).
While the stress-induced increase in ADS appears to be gen-
eral across strains and stress types, the assumed increase in
mean doubling time hTi is not. This metric actually decreases
in WT E. coli under 42�C heat shock (Figures S3I–S3K), due to
increased ribosomal activity at higher temperatures (Farewell
and Neidhardt, 1998). In terms of our model, this means that
the minimum doubling time tmin, which we assumed to be
constant, is in fact environment dependent. We emphasize that
variations in tmin between different stress levels does not quali-
tatively affect our findings.
Mechanistic Explanation of Stress-Adaptive Effect ofADS from Aggregate DynamicsHaving confirmed the existence of the stress adaption (mani-
fested by increase in s), we next investigated the mechanism
by which population diversity increases with stress. Our theoret-
ical investigations predicted two different mechanisms (see
Equation 3). The first is an increase in the total inheritable dam-
age from each cell Dancestor , which is tied to the distribution
of damage within the population and thus a signature of popu-
lation-level response. Second, ADS can also be increased
by changes in single-cell response, if the asymmetry param-
eter, a, changes with stress. By distinguishing between the
two, one can gain significant insight into mechanisms behind
observed ADS.
Consistent with the first mechanism, we found that higher
stress was associated with larger amounts of damage, as indi-
cated by the increased frequency and size of fluorescently
labeled protein aggregates (see Figures 4B, 4E, S4A, and
S4B). However, it was not clear whether this was due to
increased aggregate inheritance or solely due to an increased
rate of new aggregate formation. Upon closer investigation,
we observed that the aggregates became more stable under
stress, with half-lives larger than cell doubling times (Figures
4C and S4C). This illustrates that damage inheritance—the
fundamental mechanism of ADS—does increase with stress.
Moreover, the rate of formation of new aggregates also
increased (Figure 4D), and formation occurred equally at each
cell pole (Figure S4D), as reported earlier (Lindner et al.,
2008). Notably, the frequency of new aggregate formation
decreased (Figure S4E), while the probability of finding an
aggregate at the old pole increased (Figure S4F) with cell
pole age. This could be caused by fusion of misfolded proteins
to pre-existing aggregates, similar in effect to aggregate fusion
recently reported for S. pombe (Coelho et al., 2014) and
M. tuberculosis (Vaubourgeix et al., 2015). Thus, the data in
Figure 4 provide the following mechanistic explanation for
stress-adaptive benefit from ADS: higher environmental stress
increases the stability and size of aggregates, leading to larger
differences DT between sister cells at division (causing larger s)
and consequently a stronger positive feedback amplifying the
subpopulation of low-damage cells. As a result, ADS reduces
the population doubling time more at higher stress. The
increased aggregate formation rate further amplifies the pop-
ulation effect of ADS as more cells experience asymmetric
division.
Single-Cell Damage-Dependent Asymmetry at DivisionContributes to the Feedback Loop at the Single-CellLevelWe have shown that stress-adaptive effects of ADS can be ex-
plained by an increase in heritable damagewithin the population.
However, it was unclear whether this mechanism is sufficient to
account for the observed effects. We therefore tested whether
the strength of ADS at each stress level could be explained solely
by the stress parameter, l, assuming no change in the asymme-
try parameter, a. We used the difference in doubling times DT of
the extreme lineages to fit our model (the damage-accumulating
and damage-evading lineages highlighted by thick magenta and
green lines in Figure 1A, Supplemental Experimental Proce-
dures, III.6). The model successfully captured most of the fea-
tures in DT (dashed lines in Figure 5A) and passed two validation
criteria. First, it reproduced the increased difference in doubling
times at higher stress. Second, it captured the near-linear
Cell Systems 3, 187–198, August 24, 2016 191
A
C
D
B
Figure 3. ADS Signatures at Low and High Stress
(A and B) Heatmaps of pairwise differences in median doubling times at 37�C and 42�C show that differences between lineage groups are larger under heat
stress. Statistically significant entries are outlined in black.
(C and D) Noise-filtered distributions of doubling times at 37�C and 42�C. The mean and the width of the distribution are larger under heat stress.
See also Figure S3.
192 Cell Systems 3, 187–198, August 24, 2016
A
D E
B C
Figure 4. Aggregate Dynamics Provides the Mechanism of Stress-Adaptive ADS
(A) Representative bright-field image of E. coli cells (MC4100DclpB) with fluorescence image of ClpB-YFP overlaid in green; fluorescent foci mark protein ag-
gregates (Winkler et al., 2010).
(B) Fraction of cells with protein aggregates. More cells harbor aggregates at higher stress. Here and in (D) and (E), error bars indicate the SE of the frequency
caused by the finite counting statistics (see Experimental Procedures).
(C) Distribution of aggregate lifetimes. Aggregates are longer lived at higher stress, with very few aggregates persisting longer than a mean doubling time of
�27 min under low stress.
(D) Fraction of cells with newly formed aggregates. Aggregates form at a higher rate under stress.
(E) Frequency of aggregates per cell as a function of pole age (mean with error bars showing 95% confidence interval) is constant around 0.1 and independent of
pole age for low stress (blue) but exhibits a strong dependence on pole age and a much higher level at higher stress (red). Only pole ages where at least one
aggregate was detected have been included.
See also Figure S4.
relationship between DT at low and high stress we observed in
experimental data (Figure S5A and Supplemental Information,
section IV.6.2). However, it failed to fit the data for the early
pole ages (Figure 5A). Furthermore, the fit did not improve
when the asymmetry parameter, a, was fitted independently at
each stress level.
The relatively poor fit of the model with constant asymmetry
a suggested that the increase in stress alone is insufficient to
explain the increase in observed ADS. We therefore consid-
ered the possibility that a could have a more complex depen-
dency on stress or damage levels. To test this, we analyzed
the inheritance of aggregates as a function of pole age. If a
is constant, then the probability of aggregate inheritance
should not depend on the pole age. However, Figure 5B shows
that older-pole cells have a higher probability of retaining
aggregates following division. Contrary to the underlying
assumption of most previous studies (Erjavec et al., 2008;
Wang et al., 2010; Ackermann et al., 2007; Watve et al.,
2006; Rang et al., 2012), a is not the same for all cells in a pop-
ulation, but depends on the pole age and the amount of accu-
mulated damage in the cell. Aggregate inheritance at the
single-cell level is mostly binary because cells typically have
either zero or one aggregate (the frequency is less than 1 in
Figures 4E and S4B). However, the averaged damage-depen-
dent asymmetry measured from aggregate inheritance follows
a sigmoidal, Hill-like function (Figures 5C, S5B, and S5C),
described by Equation 4 (constant c3 corresponds to hill coef-
ficient, c1 and c2 control the offset and the inflection point,
best-fit values are given in Figure S5C).
aðDÞ= c1 +Dc3
c2 +Dc3: (Equation 4)
The damage-dependent asymmetry is a direct conse-
quence of long-lived aggregates passively migrating to the
old pole after a few divisions, as described by Lindner et al.
(Lindner et al., 2008) (see also Figure S4F). This occurs
because aggregates, which initially form anywhere in the
cell, will get pushed toward the cell poles due to nucleoid oc-
clusion (Winkler et al., 2010; Potten et al., 2002). Furthermore,
Cell Systems 3, 187–198, August 24, 2016 193
A
C
B Figure 5. Experimental Evidence for Dam-
age-Dependent Asymmetry
(A) The difference in experimental doubling times
between the damage-accumulating and damage-
evading lineages starting from the same cell. Error
bars indicate the SEM. The dashed lines show the
fit of the ‘‘constant a’’ model, while the solid lines
show the fit of the ‘‘a(D) model’’ (see Experimental
Procedures).
(B) Frequencies of inherited aggregates f inhNP and
f inhOP of new-pole and old-pole lineages at 42C
illustrate an increasing probability of aggregate
inheritance with pole age of old-pole lineages and
near-constant frequency for new-pole lineages.
We do not plot the values for low stress, since the
aggregates did not typically persist for multiple
generations. Error bars indicate SE of the fre-
quency (see Experimental Procedures).
(C) Asymmetry measured by area of aggregates at
division plotted as a function of the aggregate area
of ancestor.
See also Table S2 and Figure S5. Error bars
indicate SE.
the non-linearity of the stress dependent asymmetry may
result from a combined effect of larger aggregates diffusing
slower (Potten et al., 2002), as well as recently reported
fluid-to-glass like transition of the bacterial cytoplasm (Parry
et al., 2014) contributing to further immobilization of the
larger aggregates. Thus, damage-dependent asymmetry is
a single-cell emergent property arising � without any cellular
regulation � when an aggregate is inherited sufficiently
many times.
To confirm that single-cell damage-dependent asymmetry
indeed improves the model, we included this feature (Equa-
tion 4 with parameters as in Figure S5C) and refitted to the
experimental data of DT (Figure 5A, full lines). We found that
these fits were statistically significantly better under high
stress compared to the previous attempt (dashed lines). The
goodness of the fit and the statistical significance was quan-
tified according to the Akaike model selection criteria – a
statistical aid when choosing between the competing models
(Burnham and Anderson, 2002) (Supplemental Information,
section III.9; Table S2). The discrepancy between the adaptive
asymmetry a(D) model and the experimental data at one to two
consecutive accumulation/evasion events likely arise because
a cannot be measured for small aggregates below the detec-
tion limit. In the simulations, we have set a(D) to follow Equa-
tion 4. for all aggregate sizes. However, the actual asymme-
tries may be lower for small aggregates, e.g., due to their
shorter half-life and faster diffusion. Also, at low aggregate
numbers, stochastic effects not included in our model might
cause discrepancies.
194 Cell Systems 3, 187–198, August 24, 2016
The adaptive asymmetry aðDÞ model
also provided a better fit for both the
MC4100DclpB + ClpB-YFP and WT
strain under both stress types (Fig-
ure S5D; Table S2). Both the mechanistic
insights in Figure 4 and the evidence that
aggregates tend to merge (reported in
E. coli (Lindner et al., 2008) and fission yeast (Coelho et al.,
2014)) suggest that the damage-dependent asymmetry may
not be limited to E. coli.
Population-Level Benefits fromADSPersist under NoiseThe significant ADS-unrelated cell-to-cell variations in doubling
times (s42Cnoise = 6:8 min, see Figure S5E) compared to the effect
of ADS (s42CADS = 2:1 min) casts doubt on the relevance of ADS.
We explore the effect of this noise, and found that the benefit
of ADS is still significant. First, while the overall effect of
ADS on population diversity, s, is comparable to the noise,
ADS consistently showed a larger response to stress (1.8-fold
compared to 1.3-fold for coefficient-of-variance, see Supple-
mental Information, section III.10; Figures S3A, S3F, S3L, and
S3P). Second, simulations incorporating physiological levels of
noise into our experimentally validated model indicated that
the benefits of ADS, in terms of mean doubling time and popula-
tion damage, are significant even in the presence of this large
noise (see Figure 6A). The benefits of ADS also increased with
stress level in our simulations, just as our noise-free theory pre-
dicted (Figure 1C).
The effects of ADS on doubling times weremasked by noise at
small population sizes in the simulations (Figure 6A, n� 9). How-
ever, differences in damage inheritance were amplified along
the lineages and were detectable after a few generations (Fig-
ure 6A, n � 5,000). Therefore, the population-level benefits of
ADS emerge for biologically reasonable population sizes, even
when ADS-unrelated growth variance (Figure S5E) dominates
at the level of sister pairs.
A B Figure 6. Population and Single-Cell Feed-
backs Ensure Population Benefit from ADS
(A) Estimates of the population benefits from
ADS when accounting for ADS-unrelated cell-to-
cell variations (model simulations with parameters
derived from experimental data). The population
benefits are significantly different for large popu-
lation of nz5; 000 cells but not for small pop-
ulations of nz9 cells. The error bars indicate the
SE across 103 repeat simulations. Top: doubling-
time benefit, BhTi, taken as fold decrease in mean
doubling time in presence of ADS as compared to
no ADS. Bottom: damage benefit, BhDi, taken as
fold decrease in mean damage levels (see Exper-
imental Procedures for exact expressions). **Sig-
nificant difference of the mean (t test, p < 0.01),
*(t test, p < 0.05), ns, not significant (t test,
p > 0.05).
(B) Summary of the single-cell and population-
level stress-adaptive ADS. Stress increases the
difference between sister cells, Told � Tnew, either
directly on whole-population level by increasing
the amount of accumulated damage, Dancestor ,
or indirectly, on a single-cell level, by damage-
dependent asymmetry at division, aðDÞ. On a
population level, the difference in doubling times
sister inherit less damage and therefore divide more often, result-
ing in an over-representation of low-damage cells. As a result,
there is a positive feedback that acts on low-damage cells within
a population. The stress-adaptation arises from acceleration of
the positive feedback under high stress, due to an increase in
inheritable damage Dancestor . The inheritable damage is the dam-
age that escaped the repair. The increased amounts of Dancestor ,
due to increased rateofaccumulationandstability (Figure4;Clegg
et al., 2014), reflect the limitations of the repair mechanisms.
Adaptation is further improved via damage-dependency of the
asymmetry parameter a. The extent to which this asymmetry is
due to binary effects (e.g., frequency of aggregate inheritance)
or graded effects (e.g., size of inherited aggregates) is still un-
clear. Nevertheless, the similarity of aggregate dynamics under
both heat and antibiotic stresses (Figures S4A–S4C) suggest
that aggregate formation and inheritance is functionally similar
across different stresses. These results in combination with
research on S. pombe (Coelho et al., 2014), indicate that dam-
age-dependent asymmetry is likely a general feature of ADS
and is not specific to a specific type of cell.
Dynamic Picture of Damage-Distribution QuestionsPrevious Analyses of ADSOur results provide a new dynamic picture of how damage is
distributed within a population, with redistribution at each cell
division that imparts structure to the population as a whole.
This picture stands in contrast with earlier studies. Wang et al.
(Wang et al., 2010) suggested that the entire population con-
verges toward the fixed doubling time because the doubling
time of aging cells converges toward such a finite value. The
work by Rang et al. (Rang et al., 2011) reinterpreted their data
and proposed that cells converge to two steady states, one
with a doubling time of a fully rejuvenated cell, often likened to
a gamete cell, and the other of the most ‘‘aged’’ cell within the
population (the somatic cell). A consequence of this simplifica-
tion is that every division can be thought of as producing one
of each of these cell types, and the intermediate cell types could
be ignored.While the somatic cell- germ cell paradigm is a useful
conceptual tool, we found that seemingly binary events such as
single-aggregate inheritance (Figure 4E) do not result in two
distinct subpopulations of cells. Since the size of the aggregates
increases within each generation (Figure S4A) the changes in
damage and division times across lineages are graded, not
binary. So while a few extreme branches (consecutively accu-
mulating/evading lineages; referred to as new- and old-pole
attractors by Rang et al., 2011) eventually converge to fixed
Cell Systems 3, 187–198, August 24, 2016 195
damage levels (indicated by doubling times in Figure S3B and re-
ported in Wang et al., 2010), most of the lineages are constantly
moving in between these two extremes according to their
sequence of damage-accumulation and damage-evasion
events (e.g., Figures 3C and 3D). Nonetheless, the population
is in a dynamic equilibrium where the overall damage distribution
remains constant in time.
Another reason for examining population structure when
quantifying ADS is that metrics such as cell doubling time are
environment and strain dependent. This observation has an
important consequence for the choice of a right ADS measure:
while the change in tmin between strains and environments
does not affect distribution widths s as it is based on difference
in doubling times (see Equation 3), it does perturb the ratio
of sister doubling times – a commonly used metric of ADS.
(For illustration, consider the case of full asymmetry, a = 1
when newly acquired damage has a small contribution to the
doubling time, ml � tmin. The ratio of sister doubling times
Told=Tnew = ðtmin +mDancestorÞ=tmin is strongly affected by tmin).
Population-Level Effects of ADS May Apply MoreGenerally: Predictions and Implications for ObservedAsymmetric DNA and Aggregate Segregation in StemCellsWhile this work has focused on ADS in E. coli, our framework and
predictions are likely to apply more generally to any self-ampli-
fying heritable traits, including any cell population subject to
macromolecular damage. A particularly intriguing candidate is
the Immortal Strand Hypothesis (ISH) for adult stem cell popula-
tions (or alternatively, the Silent Sister Hypothesis). ISH predicts
that stem cells will minimize the accumulation of errors within
their DNA through retention of template DNA following division.
The replicated DNA, which will incur replicative errors, is passed
on to a less multipotent, or terminally differentiated, daughter
cell. As a result, the total number of mutations within all cells is
minimized. Our mathematical framework can be used to track
the accumulation of mutations through each lineage. l would
correspond to the frequency of mutations due to replicative
errors, and tmin would correspond to spontaneous mutation of
non-replicating DNA. In this case, it is not the doubling time
of each cell, but the length of each lineage that would be dam-
age- dependent. High-damage lineages would consist of fewer
cell-doubling events, as per ISH these cells are more likely to
differentiate and exit the stem-cell pool. We would expect that
the ISH in this model will result in a reduced mutational load in
a population of adult stem cells.
Although the analogy between the protein aggregates and
mutational frequency has a number of limitations and there are
many experimental difficulties in using a lineage-based approach
to test the ISH, such as incomplete knowledge of differentiation
hierarchies or lack of effective DNA probes, we urge caution in
drawing conclusions from cell-pairs alone. We expect that, as
in the case of ADS in E. coli, strand retention does not create
two distinct populations of cells – the mutation-free DNA of
stem cells, and the mutated DNA of differentiated cells. Strand
retention will impose a much more graded change in mutational
frequency across all cells, requiring a population-level analysis.
When applied to DNA and aggregate segregation in stem cells,
the lineage-based analysis may resolve ADS effects that are
196 Cell Systems 3, 187–198, August 24, 2016
currently below detection limit and thus reveal why ADSmanifes-
tation is environment and cell type-specific (Bufalino et al., 2013;
Rujano et al., 2006; Yennek et al., 2014; Conboy et al., 2007).
EXPERIMENTAL PROCEDURES
Cell Lines and Fluorescent Reporter
We performed experiments with the E. coli strain MC4100 (Smith, 2005) and a
deletion mutant for clpB, which encodes a protein involved in disassembly of
aggregates of misfolded proteins (Winkler et al., 2010). To visualize protein
aggregates, mutant cells were transformed with a plasmid containing YFP-
tagged ClpB (pHSGClpB-YFP) under the lac promoter together with a plasmid
containing constitutively expressed lacI (pBB529).The reporter was induced
by supplying IPTG (200 mM). Our ‘‘wild-type’’ cells were MC4100 transformed
with the pBB529 plasmid for selection purposes grown without the inducer.
Plasmids and the MC4100DclpbB strain were kindly donated by Juliane Win-
kler. See Supplemental Experimental Procedures for details.
Experiments, Microscopy, and Image Analysis
Freshly inoculated cells were grown overnight in a heat-controlled shaking
incubator under the prescribed stress level. For the experiments, exponen-
tial-phase cells were plated on freshlymade agarose (1.5%, including inducers
and antibiotic stressors when appropriate), which was placed directly on a mi-
croscope coverslip and sealed from the external environment using a custom
setup, enabling stable experimental conditions over extended periods of time.
We imaged the cells using 1003 objective every 3–4 min in bright field and
every �15 min in fluorescence (only mutant cells) on a temperature-controlled
microscope. The images were analyzed using custom image analysis soft-
ware. Image analysis provided single-cell growth rates and aggregate dy-
namics together with pedigree. Single-cell growth rates g were obtained
from experimental measurements of single-cell length as a function of time
[ðtÞ by fitting to exponential function [ðtÞ= [0egt . Filters were applied to ignore
cells with too few or corrupted data points. Single-cell doubling times T were
computed from accepted growth rates as
T =ln 2
g:
Aggregates were accepted if their average fluorescence intensity was at
least 1.6 times the average cell intensity. Each experiment yielded eight to
14 independent colonies and was repeated at least once on a separate day us-
ing freshly inoculated cells. See Supplemental Experimental Procedures and
Table S1, summarizing all reported experiments, for details.
Statistical Analyses
The width of the distribution of doubling times is s=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihðT � hTiÞ2i
q. For
normally distributed data, this definition is equivalent to the SD. However,
we specifically refrain from using this terminology since most of our
distributions are skewed and highly non-normal.
Averages
Population averages of some quantity X are denoted by angle brackets ðhXiÞ,and averages over all cells in a lineage group (defined above) are denoted by
an overline, X. The lineage-group averages are used to filter away non-ADS-
related noise in the experimental data. In Figure 5A, multiple lineage pairs
were analyzed per micro-colony.
The Error Introduced by Lineage Grouping
The histories of lineages pooled together in the same lineage group are similar,
but not identical. This results in a small variation in doubling times among
lineages within the same group even in the absence of noise. By running com-
puter simulations at different values of asymmetry and stress, we find this vari-
ation, quantified by coefficient of variation (CV), is below 2% (with only two bins
above 1%) and is considerably smaller than the variations introduced by noise
(CV �20%) or typical significant changes between bins across our experi-
ments (15%–50%).
Aggregate Statistics
Very few cells were observed to have more than one aggregate, and under
low-stress conditions many cells had none. Thus, we analyzed aggregate
presence/absence at the single-cell level under the approximating assumption
that it follows a binomial distribution using the corresponding standard
formulae: at any pole age, j, for the case of Njagg aggregates and Nj
cells cells,
the fraction of cells with aggregates is fj = ðNjagg=N
jcellsÞ and the SE of the
frequency sfj =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðfjð1� fjÞ=Nj
cellsÞq
.
Significant Differences of Mean Doubling Times from Different
Søren Vedel, Harry Nunns, Andrej Ko�smrlj, Szabolcs Semsey, and Ala Trusina
I. CELL CULTURE AND TIME-LAPSE MICROSCOPY
Cells were inoculated from the LB agarose plate in M63 minimal media with selection markers and were grownovernight at the specified temperature in 10 mL tubes in a shaking incubator. Cells were diluted into fresh media(with selection markers etc.) in the morning and left for ∼ 2.5 h in the incubator to reach the exponential phase.Experiments were commenced following another dilution.
Low-density cells were transferred to a 1.5% agarose pad (containing all relevant selection markers, antibiotics etc.in same concentrations as in the liquid culture), which was inverted and placed directly on a microscope cover slip(No. 1.5) and sealed with an air-tight top made from a glass cover and PDMS gasket to form a transparent containerwith cells growing between the cover slip and the agarose. All the components were kept at the target temperatureat all times. All experiments were conducted on a Nikon Eclipse Ti microscope fitted with a feedback-controlled heatcage (OkoLab H201 bold line cage incubator), Sutter Instruments Lambda SC shutter and Andor NEO SCC-1758camera using a Nikon Plan Apo 100X (DIC N2) oil objective, and images acquired using the manufacturer-suppliedNIS software.
For each experiment we identified 8-14 isolated single cells on the agar pad, each giving rise to a single colony. Eachcolony was imaged every 3-4 min in bright field (3.5 V lamp, ∼ 60 ms exposure) and every 15 min in YFP fluorescencechannel (500 ms exposure). Previous studies have used fluorescence for single-cell segmentation and tracking [1–4].However, exciting fluorescent markers in cells is known to stress them, so we opted for the less stressing brightfieldmicroscopy. Image segmentation and cell-tracking are performed on brightfield images taken slightly out of focus (∼half cell diameter) at every time step. We ensured that the resulting cell boundaries are in agreement with in-focusimages; fluorescence images are taken in focus.
Mutant: Heat shock stressStrain and plasmids Environmental stress # of independent exp. # of colonies # of accepted cells
* MC4100∆clpB w/ClpB-YFP and lacIq 37 ◦C 2 10 1756
MC4100∆clpB w/ClpB-YFP and lacIq 42 ◦C 2 11 1704
Mutant: Antibiotic stressStrain and plasmids Environmental stress # of independent exp. # of colonies # of accepted cells
* MC4100∆clpB w/ClpB-YFP and lacIq 37 ◦C + 0.0 µg/mL Km 2 10 1756
MC4100∆clpB w/ClpB-YFP and lacIq 37 ◦C + 0.5 µg/mL Km 2 11 1804
Wild-type: Heat shock stress
Strain and plasmids Environmental stress # of independent exp. # of colonies # of accepted cells
** MC4100 w/lacIq 37 ◦C 2 10 2090
MC4100 w/lacIq 42 ◦C 2 13 2084
Wild-type: Antibiotic stress
Strain and plasmids Environmental stress # of independent exp. # of colonies # of accepted cells
** MC4100 w/lacIq 37 ◦C + 0.0 µg/mL Km 2 10 2090
MC4100 w/lacIq 37 ◦C + 0.5 µg/mL Km 3 16 1948
*This is the same data used for reference in both mutant studies.**This is the same data used for reference in both wild-type studies.
TABLE S1. Summary of all Experiments Reported in this Study, Related to Experimental Procedures. We simultaneouslytracked 8-14 independent and non-interacting colonies in each experiment as described in Supplemental Experimental Proce-dures, Sec. I.
II. CHEMICALS AND REAGENTS
All reagents were purchased from Sigma-Aldrich except for agarose and IPTG, which was purchased from Serva(Agarose SERVA). Reagents were dissolved in either ethanol (chloramphenicol) or DI water (all other reagents) andfilter-sterilized prior to use.
III. STATISTICAL ANALYSIS
III.1. Statistical significance of different doubling times between lineage-groups
Supplementary to Figures 2C and 3A,B and Figures S3C,G,H,M. Since not all lineage-groups exhibit normallydistributed doubling times, we used non-parametric statistics, and to furthermore avoid artifacts of low sample sizes,we used both the two-sample Kolmogorov–Smirnov test and the Wilcoxon rank-sum/Mann–Whitney U-test with 5%significance level. As shown by black outline in main Figure 3A,B and Figures S3C,G,H,M, we do observe significantdifference for groups which are separated further in the population tree. For groups with little separation, the within-group noise is comparable with the between-group variation. We note that significantly increasing the number of cellsin our experiments will lead to statistically significant differences between a larger set of groups, as is illustrated bythe extensive data set by Stewart et al. [1] (see Figure S2).
III.2. Generating null-distributions for populations of lineage-grouped doubling times
Supplementary to Figures 2D, and 3C,D and S3D,E,I,K N,O. We generated the non-ADS null distributions for thelineage-grouped data sets as follows. Using the unfiltered (including within-group variations) lineage-sorted data, werandomly shuffled two cells between groups and repeated this process 100 times the total number of cells in all groupsto get good mixing while preserving the number of cells in each group. After shuffling the median of each groupwas computed and stored, and the process repeated over 500 times, each time starting from the original, unfilteredlineage-sorted data set. The 500 distributions of medians from each reshuffling process were pooled to generate thepresented null distribution. The point of repeating 500 times is to get an idea of how much the data shifts around asa consequence of a reshuffling event, which relates to the variance of the null distribution.
III.3. Investigation of statistically significant difference of population diversities σ
Supplementary to Figures S3A,F,L,P. To test if the population diversities (equivalent to widths of distributions ofdoubling times), σ, are significantly different at low and high stress we use bootstrapping to estimate the statisticalsignificance of the difference
∆σ = σ42 ◦C − σ37 ◦C (S1)
of the population diversities σ42 ◦C, σ37 ◦C at 42 ◦C and 37 ◦C respectively. To account for the fact that the associatednull distributions change between the two temperatures, see Figure 3C in the main text. we performed the followingstatistical analysis
Step (i) perform bootstrapping (random sampling with replacement) on lineage grouped data at 37 ◦C (using thesame number of data points as the original data set) to obtain a value for σ37 ◦C.
Step (ii) perform the same analysis on the lineage-grouped data at 42 ◦C to obtain a value for σ42 ◦C.
Step (iii) compute ∆σ from Eq. (S1).
Step (vi) repeat steps (i)− (iii) 104 times to obtain distribution of differences ∆σLS.
Step (v) repeat steps (i)− (iv) using the associated null-distribution data instead of the lineage-grouped data ateach temperature to obtain distribution of differences ∆σnull.
The null-distribution difference ∆σnull reflects the variation in sample-size in lineage groups between the two tem-peratures. The population diversities at low and high stress are statistically significantly different, if the distribu-tion of actual differences ∆σLS is significantly different from the distribution of differences in corresponding nulldistributions∆σnull. This is indeed the case as shown in Figure S3A,F,L,P. The distributions under all stresses aresignificantly different (P<0.01, two-sample t-test analysis). Thus the population diversity σ increase significantlyunder heat and kanamycin stress in both wild-type (MC4100) and MC4100∆ClpB+ ClpB-YFP strains.
III.4. Aggregate data statistics
Supplementary to Figures 4 and S4. At any pole age, the aggregate data follows a binomial distribution with anage-specific probability pa of having an aggregate. For the case with Na
agg aggregates and Nacells cells we compute
the frequency (estimated probability) of aggregates fa =Na
agg
Nacells
, and the standard deviation of the frequency σfa =√fa(1−fa)Ncells
using the standard formulae for the binomial distribution.
III.5. Standard error of the mean for populations of cells
Supplementary to Figures 5A and S5D. Distributions of single-cell doubling times were not normal, so we usedbootstrapping to obtain standard error of the mean (SEM): we resampled the data with replacement to obtain asingle realization of the population mean, and repeat a total of 104 times, thereby producing a distribution of means.This resulting distribution of means follows a normal distribution and is thus well-characterized by its standarddeviation, which is the SEM.
III.6. Difference of doubling times ∆T of damage accumulating/evading lineages
Supplementary to Figure 5A, S5A and S5D. In Figures 5A and S5D we show the difference in doubling timesbetween the consecutively damage-accumulating and consecutively damage-evading lineages (corresponding to all-green and all-magenta lineages in Figure 2A) originating from the same ancestor. We always start from an ancestorwhich has once undergone damage-evading event and at each generation record doubling times of consecutivelydamage-accumulating cells. This produces a sequence of doubling times Taccum(n) as a function of generation numbern following the initial cell for damage-accumulating. Repeating the analysis for the consecutively damage-evadinglineage produces the sequence Tevad. The difference ∆T is at each generation n the difference between the twosequences, ∆T (n) = Taccum(n)− Tevad(n).
The repeats in the experimental data are averaged at each generation n before subtracting; we use the to denoteaverage within each generation of such a lineage, our experimental results are expressed as ∆T (n) = T accum(n) −T evad(n).
III.6.1. Analytical expression
Experimentally, cells were grown at some temperature for ∼ 24 h prior to the start of an experiment, and they havetherefore adapted to this state. Since the state of the initial cell in an experiment is unknown, we take it to be the
average 〈D〉. Since 〈D〉 scales linearly with λ we will also write λ ˜〈D〉 = 〈D〉. Damage-accumulating lineages inheritthe fraction (1 + a)/2 of ancestor damage, while damage-evading inherit (1− a)/2 of ancestor damage.
Starting from the average damage 〈D〉, the damage in the damage-accumulating sibling in the first few generationsis
Gen. 0 : λ ˜〈D〉,
Gen. 1 : λ
[1 +
1 + a
2˜〈D〉],
Gen. 2 : λ
[1 +
1 + a
2
[1 +
1 + a
2˜〈D〉]]
= λ
[1 +
1 + a
2+
(1 + a
2
)2
˜〈D〉
],
(S2)
Thus, the damage in the consecutive old-pole lineage at generation q after the common ancestor is given by
Hold(q) = λ
[q−1∑i=0
(1 + a
2
)i+
(1 + a
2
)q˜〈D〉
]for q > 0,
Similarly, we find for the new-pole history starting from average damage 〈D〉
Hnew(q) = λ
[q−1∑i=0
(1− a
2
)i+
(1− a
2
)q˜〈D〉
]for q > 0.
The difference ∆T (q) of doubling time histories (old pole minus new pole) at generation q following the commonancestor is therefore given by
∆T (q) = tmin + µHold(q)− [tmin + µHnew(q)]
= µλ
([q−1∑i=0
(1 + a
2
)i+
(1 + a
2
)q˜〈D〉
]−
[q−1∑i=0
(1− a
2
)i+
(1− a
2
)q˜〈D〉
]), (S3)
which holds for q > 0.This formula (Eq. (S3)) was fitted to the experimental data in Figure 5A and S5D.
III.6.2. Linear scaling of ∆T (n) under different stresses
Equation S3 predicts that the difference ∆T (n) will follow almost the same functional form under different stresses,since the stress-dependence of 〈D〉 is secondary compared to the linear dependence of the prefactor. Higher stresses(larger λ) shift the the curve up, so plotting against each other ∆T (n) under two different stresses 1 and 2, the slopeα of the curve will be well approximated by the ratio of their stress levels α = λ2/λ1. This prediction is confirmedfrom our experimental data in Figure S5A.
III.7. Fitting population model to experimental data for the doubling-time difference ofdamage-accumulating/evading lineages ∆T = T accum − T evad
We fit our population model using both constant asymmetry and damage-dependent asymmetry to the experimentalpopulation data for the difference of doubling times of the damage-accumulating/evading lineages. For the analyticalmodel we use Eq. (S3), while we rely on simulations for the model with damage-dependent asymmetry.
We fit simultaneously to the data under low and high stress, because both can be used to infer system-specificparameter values, while each independently can be used to infer the specific stress-level. These fits serve the dualpurpose of (i) determining which of the two models best describe the data (i.e. as a test of whether damage-dependentasymmetry indeed is detectable at the population level), and (ii) determine the remaining free model parametersµ, and λ under both low and high stress, henceforth denoted λL and λH. The minimum doubling time tmin is notknow directly from our measurements, and the difference of doubling times ∆T (n) depends only very weakly on theexact value (tested the plausible range tmin ∈ [15; 30] min). We set tmin = 22 min (corresponding to the low stressconditions in our data) in all fitting procedures.
We have fitted the experimental data using the analytical model with the value a = c1 = 0.7411 obtained abovefrom the aggregate inheritance data (see Table S2), the theoretical optimum at a = 1, and using damage-dependentasymmetry with parameters from Table S2. We find that model with damage-dependent asymmetry is by far superior,indicating that experiments support only a(D) (Akaike weight w ≈ 0.99) and exclude models with constant asymmetry.
III.8. Analyses of the damage-dependent asymmetry
Using the experimental data on aggregate area asymmetry at cell division (Figure S5B), we can estimate theasymmetry function a = a(D) that this data implies, see Figure S5C. The asymmetry is clearly not constant, but hassigmoidal dependence on aggregate area. Sigmoidal functions assures that asymmetry will lie between physiologicallyplausible values 0 and 1. We find that the best fitting function is a = c1+D
c3
c2+Dc3, with parameters shown in Figure S5C.
The parameter values obtained from least-squares fitting.
III.9. Calculation of AICc values in the Akaike Information Criterion
Formally, the value of AICc is given by
AICc = −2 log(L(θ|y)
)+ 2K +
2K(K + 1)
n−K − 1, (S4)
where θ is a vector containing the model parameters (the model-specific ci’s and the unknown “noise” variance σof the residuals) maximizing the likelihood function given the experimental data y, L is the maximized likelihood
6
Condition Model λL [µm2] (37C, no KM) λH [µm2] µ [min µm−2] χ2red Akaike weight, w
Heat stress Const. asym. a = 1 same as above 0.1905 same as above 3.0803 3.290 × 10−3
Dam. dep. asym. 0.1417 1.0000 0.9966
TABLE S2. Supporting table for Figure 5A and Figure S5 showing best-fit parameters λL, λH and µ for the three models:“Const. asym. a = 0.7411“, “Const. asym. a = 1“ and “Dam. dep. asym.“ with tmin = 22 min. The low-stress parameter, λL,corresponds to the condition of 37C and no Kanamycin, thus we use same values of λL for the same strain. Stress parameteris expressed in units of aggregate size. Last two columns show the goodness of the fit: the reduced χ2-values (χ2
red) for eachmodel indicating how well it fits the experimental data relative to the uncertainty of the experimental measurement and theAkaike weights w which estimates the relative level of support in favor of each model best describing the data relative to thetwo other models. The best model (highlighted in boldface) is damage-dependent asymmetry according to the Akaike weights,while the two other models are doing substantially worse. The residuals of this best model is on par with the measurementerror and are consequently as good as can be expected.
function (likelihood function evaluated at its maximum point, i.e. at the parameter values θ) of the model given theexperimental data y, n is the sample size and K is the number of free parameters in the model together with theunknown noise variance of the residuals (that is, K = K + 1 where K is the number of free parameters in the model,e.g. the number of cis). The last term on the right hand side of this equation is the second-order correction termneeded to provide better estimates at small sample sizes.
We used least-squares fitting to obtain the optimal parameters. For our models, the least-squares optimal parametersare related to the maximum-likelihood optimal parameters through a simple relationship, assuming the residuals followa normal distribution [5]. This relationship stipulates
log(L(θ|y)
)≈ −n
2log σ2 (S5)
where σ2 =∑i ε
2i /n is given in terms of the squared residuals εi from the least-squares fit and the number of samples
n. The approximation in Eq. (S5) consists of dropping constant terms which are the same for all models, whichbecome irrelevant since only the differences in AICc values between models is of importance. We therefore replaceEq. (S4) by
AICc = n log(σ2)
+ 2K +2K(K + 1)
n−K − 1, (S6)
and use this to compute the AICc values. We have checked that using actual maximum likelihood estimation providesnumerically indentical results for the ci’s, ∆ and w for all three models, with irrelevant differences in raw AICc valuesbetween the two approaches a consequence of the approximation in Eq. (S5).
III.10. Quantification of the ADS-unrelated cell-to-cell variation
The statistical similarity of the within-age-group noise at each temperature allowed us to pool all noise fromeach age group. Interestingly, the resulting distributions, given as full lines in Figure S5, appear fairly similar.However, the fold-increase in noise distribution width σn is 1.37 between the two temperatures (σn
37 ◦C = 4.97 minand σn
42 ◦C = 6.79 min). For comparison the fold-increase in the width σLS of the distribution of lineage-sorted datais 1.93 (σLS
37 ◦C = 1.06 min and σLS42 ◦C = 2.05 min). To take into account the difference in means, we computed
the coefficient-of-variance of population doubling times C = σ/〈T 〉 for lineage-sorted data and noise (in both casesnormalizing the population-mean) at the two stress levels, and found that while the parameter does increase for
the noise distributions by a factor 1.28 (from C = 0.18 to C = 0.23) this increase is larger for the lineage-sorted
data (increase by a factor 1.8; from C = 0.04 to C = 0.07). These results are conistenet across all the strains andconditions. It follows that the variations of width of whole-population-distributions of doubling times in response toincreasing stress are caused primarily by asymmetric damage segregation, and are only secondarily a consequence ofnoise increasing in response to stress as well.
IV. AGENT-BASED NUMERICAL SIMULATIONS
The models have been implemented in Matlab R2012b. In all reported cases we have started the simulations froma single cell with 0 inherited damage. Matlab code for both models will be made available per request.
To simulate the growing population, we first construct the entire population tree for a specified number of genera-tions. Next for each cell in the tree we compute and record cell’s damage and doubling time as given in Eq. 1 and 2in the main text. To extract population quantities such as population-average doubling time, population-distributionof damage etc., we make a second sweep through the population tree and query for any property of interest. Up to∼ 30 generations of cell divisions can be simulated on a laptop within reasonable time (yielding about 109 cells).
Since the cells divide a deterministic time tmin + µD following last division, and the simulations start from a singlecell, there is a transient period when cells are synchronously divide at the same time. After about 10 generations thedivisions are slowly desynchronized. The synchronous divisions affect the sampling of distributions of doubling times(one has to go beyond 30 generations to reach steady state distributions) and to avoid it we add small-amplitudezero-mean noise to tmin for 5 consecutive generations to all cells present, starting at generation 5. The amplitude ofthis noise is µλ/5 but seems to be largely irrelevant within reasonable ranges. By this process we can thus reach afaster convergence of the damage distribution without affecting the population growth rate. This approach is valid asreal doubling times are also noisy (see Figure S5E).
IV.1. Simulations with physiological levels of noise, Figure 6A
For these simulationsthe noise is added on the ADS-unrelated minimum generation time tmin, by adding a randomnumber ξ to tmin for each cell, with ξ taken from the experimentally determined zero-mean distribution of physiologicalnoise 42 ◦C shown Figure S5E.
IV.2. Mean and standard deviation of population-distribution of doubling times, Figure 1A-C and 6A
At each time point in the simulations we obtain the instantaneous means of damage D(t) and doubling time T (t) and
width
√T 2 − T 2
of the population-distribution of doubling time. We store all values starting at t = 20× (tmin +µλ),
and for each statistic D(t), T (t) or
√T 2 − T 2
obtain a mean and a standard deviation over the storage interval whichwe use for the true population-averaged values of these statistics 〈D〉, 〈T 〉 and σ. For each statistic we also estimateand show the variations (standard deviation) across the sampling interval. Results using this approach are given inFigure 1B,C,E.
V. THEORETICAL MODEL
Here we sketche out an overall description of our analytical approach and present the final formulae. The detailsof model derivation will be published elsewhere.
The whole-population effects of ADS can be analyzed analytically by mapping the problem of ADS to the Isingmodel in physics[6]. This enables us to relate microscopic details of interactions between ancestral and current cells tothe macroscopic observables on the population level, e.g. growth rate and damage distribution among cells. The Isingmodel describes the behavior of a system of interacting physical particles. Each system state is a specific combinationof individual particle states. The particle states are binary (denoted by ±1 and often referred to as spin up and spindown states), and the probability that a particle occupies a given state is affected by the states of the nearest-neighborparticles. In the one-dimensional case, this corresponds to a linear chain of particles, known as an Ising chain.
Here, we map each lineage in a growing bacterial population onto a unique system state of an Ising chain. A lineageencompasses all known ancestors of a given cell, and every lineage begins with the first cell in the bacterial colony.
Two examples of lineages are highlighted by the green and magenta lines in Figure 1A. The state (±1) of a single cellwithin a lineage dictates the amount of damage inherited from its ancestor, preserving the concept of nearest-neighborinteractions. By relating damage inheritance to the division time of a cell (Eq. 2), we find that the sequence of cellstates (±1) within a lineage determines the total time elapsed from the first cell to the most recent division in thelineage. Thus, the time spanned by a lineage is analogous to the energy of a specific Ising chain configuration. Thegoal is to determine the statistical properties of a bacterial population at time t, which is equivalent to the statisticalproperties across many Ising chains with fixed system energy E. Thus, by this analogy, statistical moments such asthe mean and standard deviation (population width) of e.g. doubling times across the biological population of cellscan be computed using the ensemble averages of the equivalent model in statistical physics. This method allows usto derive approximate analytical expressions for the mean doubling time 〈T 〉 and the width σ of the distribution ofdoubling times.
V.1. Mean and variance of population damage and doubling time
Using these approach we find the following approximate result for the mean doubling time
〈T 〉 ≈ tmin + µλ
[2 + a2λµβε∗
(27βε∗λµ− 2)a2 − 18
8
]. (S7)
We furthermore find for the width of the population-distribution of doubling times
these expressions, a is the asymmetry parameter, tmin is the minimum damage-independent single-cell doubling time,λ is the amount of acquired environmental damage per generation, and µ is the conversion factor between damageand doubling time. All expressions have been derived assuming a > 0.
The two results from Eqs. (S7) and (S8) are given in Figure 1B and E in the main text. We find good agreementwith simulation results throughout, except for the width σ for a close to 1 when λµ is large (λµ > 1), where Eq. (S8)slightly overshoots.
V.2. Special case of symmetric division, a = 0
The special case of symmetric division (a = 0) is not handled by our approximate analytical expressions, however,in this special case all cells carrying the same amount of damage and divide and average population-properties areidentical to single-cell properties; thus, we can easily provide exact analytical expressions for the population behavior.
Inheriting exactly 50 % of the ancestor’s damage, the damage in generation n of a single cell starting from0 damage at the first generation and acquiring λ damage from the environment in each generation, is given byD(n) = λ
∑nj=1
12j−1 . For large n, this damage stabilizes to a constant level D(n) = 2λ because the geometric series∑n
j=1 1/2j−1 converges to 2 for n → ∞. In this long-time limit, all cells in the population have the same amount
of damage, and so it follows directly that 〈D〉 = 2λ and σD =√〈D(n)2〉 − 〈D(n)〉2 = 0. From the relationship
between damage and doubling time Eq. (2) in the main text, we find for the distribution of doubling times
[1] E. J. Stewart, R. Madden, G. Paul, and F. Taddei, PLoS Biology 3, e45 (2005).[2] A. B. Lindner, R. Madden, A. Demarez, E. J. Stewart, and F. Taddei, Proc. Nat. Acad. Sci. USA 105, 3076 (2008).[3] P. Wang, L. Robert, J. Pelletier, W. L. Dang, F. Taddei, A. Wright, and S. Jun, Curr. Biol. 20, 1099 (2010).[4] J. Winkler, A. Seybert, L. Konig, S. Pruggnaller, U. Haselmann, V. Sourjik, M. Weiss, A. S. Frangakis, A. Mogk, and
B. Bukau, EMBO Journal 29, 910 (2010).[5] K. P. Burnham and D. R. Anderson, Model selection and multimodel inference: a practical information-theoretic approach,
2nd ed. (Springer, 2002).[6] R. J. Baxter, Exactly solved models in statistical physics (Academic Press, 1982).
Figure S1. Effects of Asymmetry 𝑎 on Model Predictions, Related to Figure 1. Errorbars on simulation
results indicate standard deviations on the presented metrics. These arise as a consequence of using a
stochastic simulation algorithm. A. Mean doubling time ⟨𝑇⟩ decreases with asymmetry 𝑎. Theoretical
results computed from Eq. S30. B. Width of population-distribution of doubling times 𝜎 increases with
𝑎, with theoretical prediction given in Eq. S8. Results shown are for the model where all cells have the
same asymmetry parameter, tmin = 1 min, λ = 1μ2 and μ = 1 min μ-2. Details of theoretical calculations
and agent-based simulations are given in Supplemental Experimental Procedures, Sections IV and V.
Figure S2. Validation of Lineage-Based Analyses Using Data from Stewart et al.[1], related to Figure 2
A. Heatmap of the pairwise differences in median doubling times for each pair of lineage-groups, with
notations as in main Figure 2C. Each colored square in the heat map color-codes the difference between
the corresponding vertical and horizontal lineage-groups. Significant differences are outlined in black.
Statistical significance was computed using the non-parametric Wilcoxon rank-sum test (Mann-
Whitney U-test) at the 5% significance level. Similar results are obtained with non-parametric two-
sample Kolmogorov-Smirnov test.
Figure S3. ADS Quantification across Strains and Stresses, Related to Figure 3.
A. The distributions of the differences 𝛥𝜎 used to statistically test the increase in population
diversity, 𝜎, under heat stress (for data presented in Figure~3C,D). The distributions are obtained as
described in Supplemental Experimental Procedures, Sec.III.3. The differences are calculated
independently for the lineage-grouped data (filled bars) and the associated null-distribution data (open
bars). The population diversity at 42℃, 𝜎42℃ , is statistically significantly different from the population
diversity at 37℃, 𝜎42℃. *** denotes statistically significant difference (P<0.01, by two-sample t-test
analysis). B. Doubling times of lineage-groups with successive rounds of damage-accumulation (or
damage-evasion) converge to steady values. We plot the experimentally obtained doubling times of the
consecutive old-pole lineage (magenta) and new-pole lineage (green) as a function of the number of
generations from the initial cell. Errorbars mark standard error of the mean obtained via bootstrapping.
C. Heatmap, obtained as in Figure 2C, for population of MC4100ΔclpB + ClpB-YFP cells stressed by
0.5 μg/ml kanamycin. The heatmap for unstressed cells is shown in main Figure 3A.
D-E. Distributions of doubling times, obtained as in Figure 2D, for populations of MC4100ΔclpB +
ClpB-YFP cells that were either not stressed (blue) or kanamycin-stressed (red).
F. The distributions of the differences 𝛥𝜎, obtained as in A. and used to statistically test the increase in
population diversity, 𝜎, under kanamycin stress for MC4100ΔclpB + ClpB-YFP population.
G-H. Heatmaps, obtained as in Figure 2C, for populations of wild-type (MC4100) cells at 37℃ and
heat-stressed at 42℃. The colorbar is the same as in M.
I-K. Distributions of doubling times, obtained as in Figure 2D, for populations of wild-type cells at
37℃ (blue) and heat-stressed at 42℃(red).
L. The distributions of the differences 𝛥𝜎, obtained as in A. and used to statistically test the increase in
population diversity, 𝜎, under heat stress.
M-P. Figures corresponding to H-L, but in response to kanamycin stress.
Figure S4. Dynamics of Protein Aggregates Provides Mechanistic Explanation of Stress-Induced ADS,
Related to Figure 4.
A. Aggregate size increases with pole age and under stress. This confirms the model assumption that
damage gradually increases through the old-pole lineage. Errorbars indicate 95% confidence interval.
B. Supporting figure to main Figure 4E, showing the frequency of aggregates in kanamycin-stressed
cells. Errorbars indicate standard deviation (see Methods).
C. Supporting figure to main Figure 4C, showing the stability of aggregates in kanamycin-stressed
cells.
D. Aggregates are equally likely to form in any of the three regions of the cell. Dots mark the mean
over all colonies and errorbars represent standard deviation, quantifying colony-colony variations.
E. Frequency of formation of new aggregates for sister cells as a function of the pole-age of their
common ancestor. Frequency at 37℃ and 42℃ is nominally constant, except for old-pole cells at 42℃
where we observe a decrease for older pole ages, corresponding to the increasing frequency of
inheritance 𝑓𝑖𝑛ℎ seen in the main Figure 5B. Errorbars indicate standard deviation (see Methods).
F. Fraction of all aggregates found in the three intracellular regions at each pole age. Aggregates
passively move to the old pole after a few divisions when aggregate inheritance is prevalent (42℃) but
cannot do this when the aggregates are short-lived and the inheritance is thus insignificant (37℃). This
confirms the earlier observations by Lindner et al. [2].
All data is sampled in the mutant strain MC4100ΔclpB + ClpB-YFP.
Figure S5. Model Validation Steps, Related to Figure 5.
A. Model and data agree on the near-linear correlation between mean doubling time difference of
consecutive damage accumulating/evading lineages under low and high stress (∆��ℎ𝑖𝑔ℎ 𝑠𝑡𝑟𝑒𝑠𝑠 ∝∆��𝑙𝑜𝑤 𝑠𝑡𝑟𝑒𝑠𝑠 , see Sec. III.6). Circles are the means and errorbars are the standard error of the means.
Dashed lines are the error-weighted best fits and full lines are 95% confidence interval.
Pearson linear correlation coefficient ρ and the associated probabilities P of having a similar value of ρ
under the null hypothesis of no correlation between the low-stress and high-stress data indicate
statistically significant (positive) linear correlation between low-stress and high-stress conditions across
all investigated strains and stresses at the 5 % level.
B. Supplementary figure to main Figure 5C. Experimental measurements of average asymmetry ⟨𝑎⟩ =
𝐴𝑂𝑃
𝑖𝑛ℎ−𝐴𝑁𝑃𝑖𝑛ℎ
𝐴𝑂𝑃𝑖𝑛ℎ+𝐴𝑁𝑃
𝑖𝑛ℎ , where 𝐴𝑂𝑃𝑖𝑛ℎ and 𝐴𝑁𝑃
𝑖𝑛ℎ is the total are of inherited aggregates to the old-pole and new-pole
offspring, respectively.
C. Lines show the best fits of the models with constant (red) and damage-dependent (blue) asymmetry.
B.-C. Errorbars mark standard deviation.
D. Supplemental to main Figure 5A. Difference ∆�� of doubling times of extreme-branch lineages in the
population tree for both low (blue) and high stress (red) obtained from experimental data (filled
circles). The parameters inferred from model fits are summarized in Table S2. Dashed lines give result
using model with constant asymmetry (𝑎 = 1) while full lines shows a superior result from model with
damage-dependent asymmetry.
E. Quantification of ADS-unrelated noise through the distribution of ``centered'' doubling times (for
MC4100ΔclpB). For each cell, the mean of the lineage-group is subtracted from the measured doubling
time: ��𝑖𝑘 = 𝑇𝑖
𝑘 − ��𝑘, where 𝑇𝑖𝑘 is the doubling time of cell i in lineage group k and ��𝑘 is the mean
